Bsums arise from Hoelder convolutions. More...

#include <Bsum.h>

Inheritance diagram for Bsum:

Public Member Functions
	Bsum (const GiNaC::ex &nc, const GiNaC::ex &llc)

void	archive (GiNaC::archive_node &node) const override

void	read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst) override

unsigned	return_type (void) const override

void	print (const GiNaC::print_context &c, unsigned level=0) const override

unsigned	precedence (void) const override

GiNaC::ex	eval () const override

GiNaC::ex	subs (const GiNaC::exmap &m, unsigned options=0) const override

virtual GiNaC::ex	convert_to_Ssum_exvector (const GiNaC::exvector &Z1) const

virtual GiNaC::ex	expand_members (int level=0) const

GiNaC::ex	get_index (void) const

GiNaC::ex	get_letter_list (void) const

unsigned	get_depth (void) const

Protected Member Functions
GiNaC::ex	eval_ncmul (const GiNaC::exvector &v) const override

GiNaC::ex	derivative (const GiNaC::symbol &s) const override

unsigned	calchash (void) const override

Protected Attributes
GiNaC::ex	n

GiNaC::ex	letter_list

Friends
GiNaC::ex	convert_Bsum_to_Ssum (const GiNaC::ex &C)

GiNaC::ex	convert_Bsum_to_Zsum (const GiNaC::ex &C)

Detailed Description

Bsums arise from Hoelder convolutions.

A Bsum is defined by

$\sum\limits_{i_1=n+1}^N \sum\limits_{i_2=i_1+1}^N ... \sum\limits_{i_k=i_{k-1}+1}^N \frac{x_1^{i_1}}{i_1^{m_1}} \frac{x_2^{i_2}}{i_2^{m_2}} ... \frac{x_k^{i_k}}{i_k^{m_k}}$

This is equivalent to

$\left( x_k^+ \right)^{m_k} \left( x_{k-1}^+ \right)^{m_{k-1}} ... \left( x_1^+ \right)^{m_1} \frac{x_k}{1-x_k} \frac{x_{k-1}x_k}{1-x_{k-1}x_k} ... \frac{x_1 ... x_k}{1-x_1 ... x_k} \left( x_1 ... x_k \right)^n$

up to terms of the form

$\left( x^+ \right)^m \frac{x}{1-x} x^N = \sum\limits_{i=N+1}^\infty \frac{x^i}{i^m}$

Member Function Documentation

◆ convert_to_Ssum_exvector()

ex convert_to_Ssum_exvector ( const GiNaC::exvector & Z1 ) const

virtual

The basic object is

$\left( \sum\limits_{j_1=1}^{n'} \sum\limits_{j_2=1}^{j_1} ... \sum\limits_{n=1}^{j_{l-1}} \frac{y_1^{j_1}}{j_1^{g_1}} ... \frac{y_l^{n}}{n^{g_l}} \right) \sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_k = i_{k-1}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_k^{i_k}}{i_k^{m_k}}$

We use

$\sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_k = i_{k-1}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_k^{i_k}}{i_k^{m_k}}$

$= (-1)^k S(n;m_1,...m_k;x_1,...,x_k) - (-1)^k S(N;m_1,...m_k;x_1,...,x_k)$

$+ (-1)^k S(N;m_2,...m_k;x_2,...,x_k) \sum\limits_{i_1=n+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ...$

$+ (-1)^k S(N;m_k;x_k) \sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_{k-1} = i_{k-2}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_{k-1}^{i_{k-1}}}{i_{k-1}^{m_{k-1}}}$

◆ eval()

ex eval ( ) const

override

No automatic simplifications

◆ eval_ncmul()

ex eval_ncmul ( const GiNaC::exvector & v ) const

overrideprotected

No automatic simplifications

◆ get_depth()

unsigned get_depth ( void ) const

inline

Returns the depth.

◆ get_index()

GiNaC::ex get_index ( void ) const

inline

Returns the upper summation limit.

◆ get_letter_list()

GiNaC::ex get_letter_list ( void ) const

inline

Returns the letter_list.

The documentation for this class was generated from the following files:

Public Member Functions

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Function Documentation

◆ convert_to_Ssum_exvector()

◆ eval()

◆ eval_ncmul()

◆ get_depth()

◆ get_index()

◆ get_letter_list()